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workshop 8

# workshop 8 - as specified The next part asks us to find the...

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Anthony Reverri Section 27 Workshop 8 In this week’s workshop, we are given a problem involving exponential growth. The model for exponential growth is given by the function , where A is area and t is time. We are asked a few things about the function, first of we need to show that the maximum rate of growth occurs anytime when without solving the equation. To do this we need to maximize the function. More specifically, find the derivative of and find the time t to prove that its critical point occurs when . Below I will show the steps for taking the derivative of A with respect to t and solving for its critical point(s). Since k is just a constant, we can just tack it on at the end. So let or more simply By the chain rule: Set the function equal to zero and solve: Note that the denominator cancels out because the function is equal to zero We can see from the steps taken above that the maximum rate of growth occurs when

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Unformatted text preview: as specified. The next part asks us to find the solution corresponding to assuming that k = 6. We are then asked to sketch its graph. To start this problem off, we need to solve a differential equation. The equation for exponential growth is given by . We need to put our function into this form, then solve for the original function. From there we can plug in the given values and solve for the asked solution. Shown below are the calculations for the steps described above. is simply so we will solve the first integral and come back to this later. Apply u-substitution: Apply Trig substitution: by where Putting it all together: k=6 Now we apply the condition *Aside: If then Now when we plug B in we get: Now we need to solve for A: …More creative algebra gives us the equation: A graph of the function looks something like:...
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workshop 8 - as specified The next part asks us to find the...

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